Questions tagged [complex-dynamics]
This tag is for questions relating to complex dynamics, study of dynamical systems defined by iteration of functions on complex number spaces. It was an area of research established by Fatou and Julia towards the beginning of the last century.
358
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Example of a buried Julia component of a transcendental meromorphic function.
We know examples of buried Julia components (Definition: A Julia component is called buried if it is not contained in the boundary of any Fatou component) for rational functions. In 1998, McMullen ...
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1
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Equivalent description of Julia set
Let $f$ be a rational map acting on the Riemann sphere $\widehat{\mathbb{C}}$. The Julia set $\mathcal{J}_f$ is the complement of the Fatou set $\mathcal{F}_f$, defined to be the union of all open ...
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Problem about finite grand orbits
I am attempting an exercise in Milnor's Dynamics in One Complex Variable which I have slightly rephrased below:
Let $f \in \mathbb{C}(z)$ be a rational function of degree $d \geq 2$. Prove: $0, \infty$...
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2
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Complex Dynamics book Recommendation
I know the basics of Complex Analysis and Topology and I would like to learn Complex Dynamics. One book I found was Beardon's Iteration of
Rational Functions. I'm not sure whether its a good book for ...
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Mandelbrot set Proof for Bounding Circle of Period 2 Bulb
I've been searching and I can't find a proof for the bounding circle of the Period 2 Bulb in a Mandelbrot set.
Its referenced quite a bit that it is a circle with radius of $\frac{1}{4}$ and a centre ...
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Finite critical points of a polynomial are preperiodic implies the Fatou set is connected and simply connected
I'm currently going through Alan Beardon's book "Iteration of Rational Functions" and I'm a little stuck on his explanation of Corollary 9.5.3. which states that "If every finite ...
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Conjugating Branch Points $f=ghg^{-1}$
Consider the function $f:=ghg^{-1}$ on $\widetilde{\mathbb{C}}$ where $g$ is a homeomorphism and $h$ is a rational map. Why is it true that branch points of $h$ are transformed by $g$ to removeable ...
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Rational Mappings of the Annulus
Suppose $R:\widetilde{\mathbb{C}} \rightarrow \widetilde{\mathbb{C}}$ where $R(A) = B$ is a rational mapping from one annulus to another. Assume that one of the components of the complement of $A$ has ...
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Extending a covering map off an annulus
Let $R$ be a map from the Riemann sphere to itself, upon which its restriction to an annulus $A$ is a covering map to another annulus $B$.
Suppose there are critical points in one of the complementary ...
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Finite Number of Periodic Points in the Julia Set
I'm working through Sullivan's proof to his no wandering theorem, and in one of his sections he claims that the set of points of lowest period in the Julia set is finite. I am struggling to see why ...
- 125
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Critical points of a rational map implies the deficiency is non-zero
I'm currently trying to show that the critical points of a rational map occur iff their deficiency is non-zero. Just for context, I'm taking the deficiency of a point under a rational map $R$ as $\...
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Herman rings for polynomials
I am reading this link on complex dynamics and in Problem 12-1 it asks the reader to prove, using the Maximum Modulus Principle, that Herman rings cannot occur for polynomials.
I have seen this ...
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Critical points of rational mappings of annuli Fatou Component
Given a rational function $R$ from the Riemann sphere to itself and an annulus fatou component $A_0$ we can create the chain $$A_0 \xrightarrow{R} A_1 \xrightarrow{R} A_2 \xrightarrow{R} ...$$
One can ...
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How to prove that this polynomial inequality region is connected?
I understand that proof that the Mandelbrot Set is connected is not easy to follow and requires mathematical tools an Engineer probably doesn't have. But I got to wondering if there was a more ...
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Non-zero critical value of a complex polynomial
Let $f$ be a complex polynomial of degree $d \geq 2$ with a simple root at $0$. Assume further that its other roots $z_{1},\cdots,z_{d-1}$ satisfy $\min_{1\leq i\leq d-1}(|z_{1}|,\cdots,|z_{d-1}|)=1$, ...
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Reference/translation request: Doaudy and Hubbard
Does anyone know where to find an english translation of
Adrien Douady, John Hubbard - Etudes dynamique des polynomes complexes I & II, Publ. Math. Orsay. (1984-85) (The Orsay notes)
It is clearly ...
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Iterating $\tanh(A x)$ and $\lim_{x \to +\infty} \tanh^{[r]}(A x) = C_r$?
Everybody who ever studied special relativity or hyperbolic trig knows this function
$$\tanh(Ax)$$
for real $0 < A < 1$
$\tanh(A x)$ has a nice attracting fixpoint at $0$ and it is asymptotic to ...
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Generalisation of Fatou-Julia theorem
In an already answered question regarding the Fatou-Julia theorem (every basin of attraction contains a critical point), I saw that a user stated:
There are some other situations where an analogous ...
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Fatou and Julia sets under the conjugation.
I want to prove the following property of Fatou and Julia set.
Let $R$ be non-constant rational map, $g$ be Moebius map, and let $S = gRg^{-1}$. Then $F(S) = g(F(R))$ and $J(S) = g(J(R))$.
I tried ...
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Rational map satisfies Lipschitz condition.
I have rational map $R$ of the Riemann sphere $\overline{\mathbb{C}}$ onto itself. Consider the spherical metric $\sigma$ on $\overline{\mathbb{C}}$, I want to show that it $R$ satisfies the Lipschitz ...
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Necessary condition for "finite-type" maps onto compact hyperbolic Riemann surfaces
Introduction
This question concerns "finite-type" maps from a Riemann surface $X$ to a compact hyperbolic Riemann surface $Y$. (Finite-type maps are roughly like branched covers, but not ...
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Inequality involving the distance to the nearest integer
i've been searching for several days trying to prove this inequality below. Let $\alpha$ an irrationnal number.
First let's write for $n,j>0$ some integers, $\sigma_j^{+}(n)=1$ if $0<\{n\alpha\}&...
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Maclaurin series for $ f(f(z)) = \exp(z) - \exp(-z) + \exp(-z/2) - \exp(-z/5)+ \exp(-z/6) - \exp(-z/9)$
I wonder about the Maclaurin series of the analytic $f(z)$ such that
$$ f(f(z)) = g(z) = \exp(z) - \exp(-z) + \exp(-z/2) - \exp(-z/5)+ \exp(-z/6) - \exp(-z/9) $$
Since zero is a fixpoint and $g'(0) = \...
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Dynamics of the map $z \mapsto z^2$
I am studying the dynamics of the map $f(z)=z^2$ on $\overline{\mathbb{C}}$, where $\overline{\mathbb{C}}$ is Riemann sphere, from the book of Alan Beardon "Iteration of rational maps" . I ...
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For a complex number on the unit circle with irrational argument z, does $z^{N!}$ ever converge?
Let $z = e^{2\pi i\alpha}$ with $\alpha \in [0, 1[$ irrational. I'm convinced the sequence $(z^{N!})_{N \in \mathbb{N}}$ has no reason to ever stabilize but I'm not sure how to prove or disprove this.
...
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Is the interior of the mandelbrot set connected?
I know that the Mandelbrot set is connected, but what about its interior? It doesn't seem intuitively like it should be, but I can't find any information online confirming this. I can think of an ...
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Oscillations in Newton's fractal
I'm working on a program that draws Newton's fractal for a given polynomial. Newton's fractal is a fractal derived from Newton's root-finding method, which given some initial guess $x_0$ and function $...
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$f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $?
While talking about tetration with my friend the following idea (re)occured.
Equation A
$$f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $$
or variations of it like the weaker
Equation B
$$f(f(f(f(z)))) = z , ...
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Persistence of periodic points
I'm currently studying some of the more deeper results in complex dynamics theory and I've come across a statement that is sometimes used in proofs seemingly without justification.
Let $X$ be a ...
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Limits of recursions like $f(n+2)=\frac{1}{f(n)} - \frac{1}{f(n+1)}$ !?
Consider the sequences
$$f(0)=1,f(1)=2$$
$$f(n+2)=\frac{1}{f(n)} - \frac{1}{f(n+1)}$$
$$g(0)=1,g(1)=2,g(2)=3,g(3)=4$$
$$g(n+4)=\frac{3}{g(n)} - \frac{3}{g(n+1)} + \frac{3}{g(n+2)} - \frac{3}{g(n+3)}$$
...
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How to prove this theorem for the number of components of a filled julia sets?
If one finite critical point of $f(z)$ escapes to infinity by iterating, then the filled-in Julia set of $f(z)$ consists of infinitely many components.
How to prove this ?
I must admit I heard this in ...
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Number of connected components for the filled julia set of $z^2 + c z^5$
For any polynomial map $f$ we can define the filled Julia $K$ to be closure of the complement of $ \Omega$ in $\mathbb{C}$ of the basin of infinity
$$\Omega = \{z \in \mathbb{C}; f^{\circ n}(z)\...
- 15.3k
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1
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Sign of $n$ th derivative of $f(x)$?
Let $f(z)$ satisfy $f(f(z)) = \operatorname{arcsinh}(z/2)$
More precisely, we construct such an $f(z)$ by using the fixpoint at $0$ and the related Koenigs function.
see : https://en.wikipedia.org/...
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Introducing undergraduate students to dynamical systems
In my department a course on dynamical systems is offered this semester. It is a course offered to third (out of four) year undergraduate students and it involves basic dynamics of real maps, ...
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Equicontinuous Family at a Point
I am reading the definition of equicontinuous family at a point from book called "Iteration of Rational Functions" by Alan F. Beardon. There it is written that equicontinuity of family at a ...
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3
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Construction of Mandelbrot Set
I am doing Master's project in Complex Dynamics. Here I want to talk particularly about the Mandelbrot set. I have studied its formation and dynamics as parameter $c$ changes (which is very hand ...
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Examples of Hyperbolic Set and J-Stable sets
I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. The following two definitions are given without any examples in ...
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Hyperbolic Set of Extended Complex Plane
I am studying one of the research article "The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets" by Mitsuhiro Shishikura. There is one section where he gave the ...
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Math function for plotting a surface that looks like crumpled paper?
Is there a way to draw a two-dimensional or three-dimensional surface that resembles this?
All I know so far is that the dynamics of paper are more complicated than they sound, does this mean I could ...
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Are there any points on the parameter plane that do not belong to any wake?
p/q-wake is the region of parameter plane enclosed by two external rays landing on the same root point on the boundary of Mandelbrot set main cardioid (period 1 hyperbolic component).
Are there any ...
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How to prove that the Douady-Hubbard conformal map from the exterior of Mandelbrot Set to exterior of unit disc is actually holomorphic?
I was reading the Orsay Notes on Exploring the Mandelbrot Set. (https://pi.math.cornell.edu/~hubbard/OrsayEnglish.pdf)
On Page 64, it is proven that the Mandelbrot Set is connected.
I understood the ...
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About a complex sequence and complex logarithms
Intro
I am studying complex logarithms and particularly the following sequence $(z_n)$ defined by:
$$z_0 \in \mathbb{C} \\
\forall n \in \mathbb{N}, z_{n+1} = \log_{\mathrm{i}} z_n$$
It involves:
$ ...
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Pierre Fatou translation articles
A friend of mine is studying complex dynamics and require some information of the papers:
Sur les equations fonctionnelles 1919 Pierre Fatou
Sur les equations fonctionnelles 1920 Pierre Fatou
The ...
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2
answers
65
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properties of the Mandelbrot set and complex dynamical system [closed]
I want to learn some knowledge about complex dynamical system, especially about the properties of Mandelbrot set, are there any literatures about this topics?
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Points $p$ in the Mandelbrot set such that $M\setminus\{p\}$ is not connected
Obviously, only the points in the boundary $p\in\partial M$ are interesting. I managed to prove a few examples:
For $p=\frac 14$ the set remains connected, since $M\cap \mathbb R=[-2,\frac 14]$ and ...
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A set analogous to the Mandelbrot set which is connected but not locally connected
Consider the sequence of polynomials $P_1(z)=z$ and
$$P_{n+1}(z)=z+P_n(z)^2.$$
We can think of the Mandelbrot set as the intersection of the images of the disk of radius 2
$$\bigcap_{n=1}^\infty P_n^{-...
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0
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Deforming roots of a cubic polynomial
This is problem 9-1 from Milnor, dynamics of one complex variable (arxiv).
Let $f_{\alpha}(z) = z + \alpha z^2 + z^3$. Show that $f_{\alpha}$ can be perturbed so that the double fixed point at the ...
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Equicontinuity is not an open condition
I am reading the book "Arithmetic Dynamics" from Joseph Silverman that has the following comment about the Fatou set:
where the Fatou of a map means the Fatou set of its iterations. I could ...
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3
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396
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Function to calculate the period of a Mandelbrot Set point
I'm trying to implement a program to calculate the period of a Mandelbrot Set point.
I'm using the multipliers of periodic points as explained here:
https://rlbenedetto.people.amherst.edu/talks/...
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2
answers
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How to find the initial point whose sequence of iterates converges {$z_n$}?
Consider the iterration of rational function $R$($z$),
Suppose the sequence of iterates {$z_n$} of initial point $z_0$ converges to $w$. Then (because $R$ is continuous at $w$),
$w$ = $\displaystyle \...
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